Date(s) : 23/02/2017 iCal
14 h 00 min - 15 h 00 min
For a real reductive group G, the center z(U(g)) of the universal enveloping algebra of the Lie algebra g of G acts on the space of distributions on G. This action proved to be very useful.
Over non-Archimedean local fields, one can replace this action by the action of the Bernstein center z of G, i.e. the center of the category of smooth representations. However, this action is not well studied. In my talk I will provide some tools to work with this action and discuss the following results.
1) The wave-front set of any z-finite distribution on G over any point x∈G lies inside the nilpotent cone of $T^∗_xG≅g$.
2) Let $H_1,H_2$⊂G be symmetric subgroups. Consider the space J of $H_1×H_2$-invariant distributions on G. We prove that the z-finite distributions in J form a dense subspace. In fact we prove this result in wider generality, where the groups H_i are spherical groups of certain type and the invariance condition is replaced by semi-invariance. Further we apply those results to density and regularity of spherical characters.
The first result can be viewed as a version of Howe’s expansion of characters. The second result can be viewed as a spherical space analog of a classical theorem on density of characters of admissible representations. It can also be viewed as a spectral version of Bernstein’s localization principle.
In the Archimedean case, the first result is well-known and the second remains open.
I will also describe an application of these results to the non-vanishing of certain spherical Bessel functions.
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